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Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface

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Abstract

For a normal variation of a hypersurface Mn in a space form Q n+1 c by a normal vector field fN, R. Reilly proved:

$$\frac{d}{{dt}}S_{r + 1} (t)|_{t = 0} = L_r f + (S_1 S_{r + 1} - (r + 2)S_{r + 2} )f + c(n - r)S_r f,$$

where L r (0 < r < n − 1) is the linearized operator of the (r + 1)-mean curvature S r+1 of Mn given by L r = div(P r ∇); that is, L r = the divergence of the rth Newton transformation P r of the second fundamental form applied to the gradient ∇, and L0 = Δ the Laplacian of Mn.

From the Dirichlet integral formula for L r

$$\int {_{M^n } } (fL_r g + \left\langle {P_r \nabla f,\nabla g} \right\rangle ) = 0$$

new integral formulas are obtained by making different choices of f and g, generalizing known formulas for the Laplacian. The method gives a systematic process for proofs and a unified treatment for some Minkowski type formulas, via L r .

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  1. Departamento de Matemática, Universidade Federal de Alagoas, 57072-970, Maceio – Al, Brazil

    Hilario Alencar & A. Gervasio Colares

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  1. Hilario Alencar
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  2. A. Gervasio Colares
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Alencar, H., Colares, A.G. Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface. Annals of Global Analysis and Geometry 16, 203–220 (1998). https://doi.org/10.1023/A:1006555603714

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